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Given a dynamical system whose description includes time-varying uncertain parameters, it is often desirable to design an output feedback controller leading to asymptotic stability of a given equilibrium point. When designing such a controller, one may consider static (i.e., memoryless) or dynamic compensation. In this paper, we show that solvability of various output feedback design problems is implied by existence of a solution to a certain constrained Lyapunov problem (CLP). The CLP can be stated in purely algebraic terms. Once the CLP is described, we provide necessary and sufficient conditions for its solution to exist. Subsequently, we consider application of the CLP to a number of robust stabilization problems involving static output feedback and observer-based feedback.